Finite Group Databasebeta

← #35226056

Subgroups — #35226056

Order 20864.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104759C211yesmaximal, characteristic, complemented
2C2×#19104756C211yesmaximal, characteristic, complemented
3#23782319C211yesmaximal, characteristic, complemented
4#23782308C211yesmaximal, characteristic, complemented
5#23782060C211yesmaximal, characteristic, complemented
6#23781922C211yesmaximal, characteristic, complemented
7#23781893C211yesmaximal, characteristic, complemented
8#23781871C211yesmaximal, characteristic, complemented
9#23781830C211yesmaximal, characteristic, complemented
10#23781824C211yesmaximal, characteristic, complemented
11#23781817C211yesmaximal, characteristic, complemented
12#23781807C211yesmaximal, characteristic, complemented
13#23781775C211yesmaximal, characteristic, complemented
14#23781735C211yesmaximal, characteristic, complemented
15#463×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579698C2211yescharacteristic
18C2×#14579698C2211yescharacteristic, complemented
19C2×D4×D163C2211yescharacteristic
20#19104759—24no—
21#19104756—24no—
22C2×#14579689C2211yescharacteristic, complemented
23C2×#14579689C2211yescharacteristic, complemented
24C2×#14579688C2211yescharacteristic
25C2×#14579688C2211yescharacteristic
26C2×#14579687C2211yescharacteristic
27C2×#14579687C2211yescharacteristic
28C2×#14579684C2211yescharacteristic
29C2×#14579682C2211yescharacteristic
30C2×C8×D163C2211yescharacteristic, complemented
31#19104735C2211yescharacteristic, complemented
32#19104719—21no—
33#19104719—21no—
34#19104717—21no—
35#19104715C2211yescharacteristic
36#19104715C2211yescharacteristic, complemented
37#19104711C2211yescharacteristic
38#19104709—21no—
39#19104708—21no—
40#19104706C2211yescharacteristic
41#19104705—21no—
42C4×D652C2211yescharacteristic
43C4×D652C2211yescharacteristic
44C2×C4⋊Dic163—21no—
45C22⋊C4×D163—21no—
46#19104666C2211yescharacteristic, complemented
47#19104660C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104656C2211yescharacteristic, complemented
50#19104655C2211yescharacteristic
51(C2×D163)⋊C8C2211yescharacteristic
52(C2×D163)⋊C8C2211yescharacteristic, complemented
53#19104646C2211yescharacteristic, complemented
54C2×SD16×C163C2211yescharacteristic, complemented
55C2×D8×C163C2211yescharacteristic
56#124×C163C2211yescharacteristic
57C4×D4×C163C2211yescharacteristic, complemented
58C4⋊C8×C163C2211yescharacteristic, complemented
59#103×C163C2211yescharacteristic, complemented
60#102×C163C2211yescharacteristic
61C23×D163—21no—
62#14579700—22no—
63#14579700—41no—
64#14579698—22no—
65#14579698—22no—
66#14579698—41no—
67#14579698—41no—
68C2×Dic326C2311yescharacteristic
69C2×Dic326C2311yescharacteristic
70C2×D652—21no—
71C2×D652D411yescharacteristic
72C2×D652C2311yescharacteristic
73C2×C22⋊D163—21no—
74C2×C22⋊D163—21no—
75C2×C22⋊D163—21no—
76C2×C22⋊D163—21no—
77D4×D163—22no—
78D4×D163—41no—
79C22×Dic163—21no—
80C22×Dic163—21no—
81C2×C4×D163—21no—
82C2×C4×D163—21no—
83C2×C4×D163C2311yescharacteristic
84C2×C4×D163C2311yescharacteristic
85#14579689—22no—
86#14579689—41no—
87#14579688—22no—
88#14579688—22no—
89#14579687—22no—
90#14579687—41no—
91#14579684D414yescomplemented
92#14579682—22no—
93C2×C163⋊C8C2311yescharacteristic
94C2×C163⋊C8C2311yescharacteristic
95C8×D163—22no—
96C4⋊Dic163—21no—
97C4⋊Dic163—21no—
98C4⋊Dic163—21no—
99C4⋊Dic163—21no—
100C4⋊Dic163D411yescharacteristic
101C4⋊Dic163C2311yescharacteristic
102Dic163⋊C4—21no—
103C22⋊Dic163—21no—
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4—21no—
106(C2×D163)⋊C4—21no—
107(C2×D163)⋊C4—21no—
108(C2×D163)⋊C4D411yescharacteristic
109(C2×D163)⋊C4D411yescharacteristic
110C2×Q8×C163C2311yescharacteristic
111C2×D4×C163C2311yescharacteristic
112C2×D4×C163C2311yescharacteristic
113C22×C4×C163—21no—
114SD16×C163—22no—
115D8×C163—22no—
116C2×C8×C163C2311yescharacteristic
117C2×C8×C163C2311yescharacteristic
118C4⋊C4×C163C2311yescharacteristic
119C22⋊C4×C163—21no—
120C22⋊C4×C163—21no—
121C22⋊C4×C163—21no—
122C42×C163C2311yescharacteristic
123C22×D163—21no—
124C22×D163—42no—
125C22×D163—42no—
126C22×D163—81no—
127C22×D163C2×D411yescharacteristic
128C22×D163C2×D411yescharacteristic
129Dic326—21no—
130Dic326—21no—
131Dic326—21no—
132Dic326—21no—
133D652—21no—
134D652—41no—
135D652—42no—
136D652C2×D412yes—
137C22⋊D163—42no—
138C22⋊D163—42no—
139C22⋊D163—42no—
140C22⋊D163—81no—
141C2×Dic163—21no—
142C2×Dic163—21no—
143C2×Dic163—21no—
144C2×Dic163—42no—
145C2×Dic163—42no—
146C2×Dic163C2×D411yescharacteristic
147C2×Dic163C2×D411yescharacteristic
148C4×D163—21no—
149C4×D163—21no—
150C4×D163—21no—
151C4×D163—21no—
152C4×D163—42no—
153C4×D163—81no—
154C163⋊C8—21no—
155C163⋊C8C2×D412yes—
156C23×C163—21no—
157C23×C163—21no—
158Q8×C163—21no—
159Q8×C163—21no—
160D4×C163—21no—
161D4×C163—21no—
162D4×C163—21no—
163D4×C163C2×D412yes—
164C2×C4×C163—21no—
165C2×C4×C163—21no—
166C2×C4×C163—42no—
167C2×C4×C163C2×D411yescharacteristic
168C2×C4×C163C2×D411yescharacteristic
169C2×C4×C163C2411yescharacteristic
170C8×C163—21no—
171C8×C163—21no—
172C22×C163—22no—
173C22×C163—41no—
174C22×C163—41no—
175C22×C163—41no—
176C22×C163—41no—
177C22×C163—41no—
178C22×C163C22×D411yescharacteristic
179C2×D163—21no—
180C2×D163—21no—
181C2×D163—22no—
182C2×D163—21no—
183C2×D163—21no—
184C2×D163—21no—
185C2×D163—21no—
186C2×D163—41no—
187C2×D163—41no—
188C2×D163—84no—
189C4×C163—21no—
190C4×C163—21no—
191C4×C163—41no—
192C4×C163—41no—
193C4×C163C22×D411yescharacteristic
194C4×C163H(5,𝔽2)11yescharacteristic
195Dic163—21no—
196Dic163—21no—
197Dic163—41no—
198Dic163—41no—
199Dic163—41no—
200C2×C163—42no—
201C2×C163—81no—
202C2×C163D4211yescharacteristic
203C2×C163#57511yescharacteristic
204C2×C163#57711yescharacteristic
205D163—41no—
206D163—41no—
207D163—42no—
208C163#326711yescharacteristic, complemented, Sylow
209#3267—1631nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1631no—
211C2×#135—1631no—
212#547—1631no—
213#546—1631no—
214#497—1631no—
215#470—1631no—
216#465—1631no—
217#463D16311yescharacteristic, complemented
218#461—1631no—
219#452—1631no—
220#452—1631no—
221#450—1631no—
222#449—1631no—
223#442—1631no—
224#435—1631no—
225C2×#41—1631no—
226C2×#41—1631no—
227C2×#41—1631no—
228C22×D4—1631no—
229AGL(1,ℤ/8)—3264no—
230#135—3264no—
231C2×Q16—1631no—
232C2×Q16—1631no—
233C2×SD16—1631no—
234C2×SD16—1631no—
235C2×SD16—1631no—
236C2×SD16—1631no—
237C2×SD16C2×D16311yescharacteristic, complemented
238C2×D8—1631no—
239C2×D8C2×D16311yescharacteristic
240C2×M16—1631no—
241C22×C8—1631no—
242C4⋊Q8—1631no—
243#124C2×D16311yescharacteristic
244#123—3261no—
245#123—3261no—
246#123—3261no—
247C22⋊Q8—1631no—
248C22⋊Q8—1631no—
249C22⋊Q8—1631no—
250#121—3261no—
251#121—3261no—
252#121—1631no—
253#120—3261no—
254C4×D4—1631no—
255C4×D4—1631no—
256C4×D4C2×D16311yescharacteristic, complemented
257C2×C4⋊C4—3261no—
258C2×C22⋊C4—3261no—
259#106—1631no—
260C4⋊C8C2×D16311yescharacteristic, complemented
261#103—1631no—
262#103—1631no—
263#103C2×D16311yescharacteristic, complemented
264#102—1631no—
265#102—1631no—
266#102C2×D16311yescharacteristic
267C22⋊C8—1631no—
268C22⋊C8—1631no—
269#97—1631no—
270C24—3261no—
271#41—3262no—
272#41—3262no—
273#41—3262no—
274#41—6521no—
275#41—6521no—
276#41—6521no—
277C2×Q8—1631no—
278C2×Q8—1631no—
279C2×Q8C22×D16311yescharacteristic
280C2×D4—3261no—
281C2×D4—3262no—
282C2×D4—3261no—
283C2×D4—3261no—
284C2×D4—3261no—
285C2×D4—3261no—
286C2×D4—6521no—
287C2×D4—1631no—
288C2×D4—1631no—
289C2×D4C22×D16311yescharacteristic
290C2×D4C22×D16311yescharacteristic
291C22×C4—21no—
292C22×C4—3261no—
293C22×C4—3261no—
294C22×C4—3261no—
295C22×C4—3261no—
296C22×C4—1631no—
297C22×C4—1631no—
298Q16—3262no—
299Q16—6521no—
300SD16—22no—
301SD16—3262no—
302SD16—3262no—
303SD16—3262no—
304SD16—6521no—
305D8—22no—
306D8—1634no—
307M16—3262no—
308C2×C8—3262no—
309C2×C8—1631no—
310C2×C8—1631no—
311C2×C8C22×D16311yescharacteristic
312C2×C8C22×D16311yescharacteristic
313C4⋊C4—3261no—
314C4⋊C4—3261no—
315C4⋊C4—3261no—
316C4⋊C4—3261no—
317C4⋊C4—3261no—
318C4⋊C4—1631no—
319C4⋊C4—1631no—
320C4⋊C4C22×D16311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—3261no—
325C22⋊C4—3261no—
326C22⋊C4—3261no—
327C22⋊C4—3261no—
328C22⋊C4—3261no—
329C22⋊C4—1631no—
330C22⋊C4—1631no—
331C42C22×D16311yescharacteristic
332C23—21no—
333C23—21no—
334C23—13041no—
335C23—3261no—
336C23—6522no—
337C23—6522no—
338C23—1631no—
339C23—1631no—
340Q8—21no—
341Q8—21no—
342Q8—3261no—
343Q8—3261no—
344Q8—3261no—
345Q8—3261no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—13041no—
350D4—3261no—
351D4—6521no—
352D4—6522no—
353D4—6522no—
354D4—6522no—
355D4—6522no—
356D4—1632no—
357D4D4×D16312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—13041no—
362C2×C4—3261no—
363C2×C4—3261no—
364C2×C4—3261no—
365C2×C4—3261no—
366C2×C4—3261no—
367C2×C4—3261no—
368C2×C4—3261no—
369C2×C4—6522no—
370C2×C4—6522no—
371C2×C4—6522no—
372C2×C4—1631no—
373C2×C4—1631no—
374C2×C4D4×D16311yescharacteristic
375C2×C4D4×D16311yescharacteristic
376C2×C4C23×D16311yescharacteristic
377C8—21no—
378C8—21no—
379C8—3261no—
380C8—1632no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—13044no—
388C22—3261no—
389C22—3261no—
390C22—3261no—
391C22—3261no—
392C22—3261no—
393C22—3261no—
394C22—3262no—
395C22—6521no—
396C22—6521no—
397C22C2×D4×D16311yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—3261no—
403C4—3261no—
404C4—6521no—
405C4—6521no—
406C4—6521no—
407C4C2×D4×D16311yescharacteristic
408C4#1910478111yescharacteristic
409C2—42no—
410C2—81no—
411C2—6521no—
412C2—6522no—
413C2—6521no—
414C2#2378229911yescharacteristic
415C2#2378246111yescharacteristic
416C2#2378247211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#463×C163C2
socleC22×C163C22×D4
hypercenter#463D163
hypocenterC163#3267
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#19104759C2semidirect
2. C2×#19104756C2semidirect
3. #23782319C2semidirect
4. #23782308C2semidirect
5. #23782060C2semidirect
6. #23781922C2semidirect
7. #23781893C2semidirect
8. #23781871C2semidirect
9. #23781830C2semidirect
10. #23781824C2semidirect
11. #23781817C2semidirect
12. #23781807C2semidirect
13. #23781775C2semidirect
14. #23781735C2semidirect
15. #463×C163C2semidirect
18. C2×#14579698C22semidirect
22. C2×#14579689C22semidirect
23. C2×#14579689C22semidirect
30. C2×C8×D163C22semidirect
31. #19104735C22semidirect
36. #19104715C22semidirect
46. #19104666C22semidirect
47. #19104660C22semidirect
48. #19104660C22semidirect
49. #19104656C22semidirect
52. (C2×D163)⋊C8C22semidirect
53. #19104646C22semidirect
54. C2×SD16×C163C22semidirect
57. C4×D4×C163C22semidirect
58. C4⋊C8×C163C22semidirect
59. #103×C163C22semidirect
91. #14579684D4semidirect
208. C163#3267semidirect
217. #463D163semidirect
237. C2×SD16C2×D163semidirect
256. C4×D4C2×D163semidirect
260. C4⋊C8C2×D163semidirect
263. #103C2×D163semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#191047592
C2×#191047562
#237823192
#237823082
#237820602
#237819222
#237818932
#237818712
#237818302
#237818242
#237818172
#237818072
#237817752
#237817352
#463×C1632
#3267163